We describe the scheme structure of the Hilbert scheme of points of affine 3-space,in terms of representations of the Jacobi algebra of a quiver with potential. This exhibits the Hilbert scheme of points as the critical locus of a regular function on a smooth variety.We discuss the torus action on the Hilbert scheme and its Euler characteristic.

We will discuss the connection between cyclotomic rational Cherednik algebras at t=0 and the Hilbert scheme of points in the plane. In particular, we will explain how the spectrum of the centre of the rational Cherednik algebra is diffeomorphic to a certain component of the Hilbert scheme. Analyzing torus actions we will derive some combinatorial applications such as Bezrukavnikov and Finkelberg's proof of Haiman's conjecture about wreath MacDonald polynomials and a generalization of the q-hook formula.

This is an overview talk about Deligne categories. These categories interpolate (representation) categories to complex parameters (or parameters in any field). Initially Deligne considered the cases $\underline{Rep}(O_t)$, $\underline{Rep}(GL_t)$, $\underline{Rep}(S_t)$ for $t \in \mathbb{C}$ which interpolate representations of the orthogonal groups, the general linear and the symmetric groups (we refer to those as classical Deligne categories).